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Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong Venkatesh Saligrama Boston University

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Page 1: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal

bounds with efficient algorithmsChun Lam Chan, Pak Hou Che and Sidharth Jaggi

The Chinese University of Hong Kong

Venkatesh Saligrama

Boston University

Page 2: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal

bounds with efficient algorithmsChun Lam Chan, Pak Hou Che and Sidharth Jaggi

The Chinese University of Hong Kong

Venkatesh Saligrama

Boston University

n-dd

Page 3: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal

bounds with efficient algorithmsChun Lam Chan, Pak Hou Che and Sidharth Jaggi

The Chinese University of Hong Kong

Venkatesh Saligrama

Boston University

n-dd

Page 4: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

4

Literature No error: [DR82], [DRR89] With small error ϵ:

Upper bound: [AS09], [SJ10]

Page 5: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

5

Literature No error: [DR82], [DRR89] With small error ϵ:

Upper bound: [AS09], [SJ10]

Lower bound: [Folklore]

Page 6: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal

bounds with efficient algorithms

Page 7: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

7

Algorithms motivated by Compressive Sensing Combinatorial Basis Pursuit (CBP) Combinatorial Orthogonal Matching Pursuit

(COMP)

Page 8: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

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Noiseless CBP

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Page 9: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

9

Noiseless CBP

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Discard

Page 10: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

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Noiseless CBP Sample g times to form a

group

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Page 11: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

11

Noiseless CBP Sample g times to form a

group

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Page 12: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

12

Noiseless CBP Sample g times to form a

group

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Page 13: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

13

Noiseless CBP Sample g times to form a

group

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Page 14: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

14

Noiseless CBP Sample g times to form a

group

Total non-defective items drawn:

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Page 15: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

15

Noiseless CBP Sample g times to form a

group

Total non-defective items drawn:

Coupon collection:

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Page 16: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

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Noiseless CBP Sample g times to form a

group

Total non-defective items drawn:

Coupon collection:

Conclusion:

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Page 17: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

17

Noisy CBP

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Page 18: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

18

Noisy CBP

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Page 19: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

19

Noisy CBP

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Page 20: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

20

Noisy CBP

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Page 21: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

21

Noiseless COMP

Page 22: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

22

Noiseless COMP

Page 23: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

23

Noiseless COMP

Page 24: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

24

Noiseless COMP

Page 25: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

25

Noiseless COMP

Page 26: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

26

Noisy COMP

Page 27: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

27

Noisy COMP

Page 28: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

28

Noisy COMP

 

Page 29: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

29

Noisy COMP

Page 30: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

30

Noisy COMP

Page 31: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

31

Noisy COMP

Page 32: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

32

Noisy COMP

Page 33: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

33

Simulations

0 100 200 300 400 500 600 700 8000

1

Experimental; q=0

Theoretical-lower; q=0

Theoretical-upper;q=0

number of tests (T)

succ

ess

rate

Page 34: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

34

Simulations

0 500 1000 1500 2000 2500 30000

1

Experimental; q=0

Experimental; q=0.1

Experimental; q=0.2

Theoretical-lower; q=0

Theoretical-lowerl; q=0.1

Theoretical-lower; q=0.2

Theoretical-upper;q=0

Theoretical-lower; q=0.1

Theoretical-lower; q=0.2

number of tests (T)

succ

ess

rate

Page 35: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

35

Summary

CBP COMP

Noiseless

Noisy

With small error ,

Page 36: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

36

End

Thanks

Page 37: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

37

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y

0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 0

0 0 1 1 0 1 1 0 0 1

Page 38: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

38

x 0 0 1 0 0 0 1 0 0

M y

0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 0

0 0 1 1 0 1 1 0 0 1

0 10 11 0 x90 1 → 00 11 00 1

Noiseless COMP

Page 39: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

39

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y

0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 0

0 0 1 1 0 1 1 0 0 1

0 01 10 0 x71 1 → 11 10 01 1

Page 40: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

40

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y

0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 0

0 0 1 1 0 1 1 0 0 1

1 11 10 0 x40 1 → 11 10 01 1

Page 41: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

41

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

1 1 0 0 0 11 1 1 1 0 10 0 x4 0 0 x7 1 0 x9

(a) 0 1 → 1 (b) 1 1 → 1 (c) 0 1 → 01 1 1 1 0 10 0 0 0 1 01 1 1 1 0 1

Page 42: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

42

Noisy COMPx 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 00 00 11 01 10 01 1

Page 43: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

43

Noisy COMPx 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 00 00 1 x31 0 → 11 10 01 1

If then =1 else =0

Page 44: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

44

Noisy COMPx 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 00 01 1 x2

1 0 → 1

1 10 00 1

Page 45: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

45

Noisy COMPx 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 01 00 1 x71 0 → 00 10 01 1

Page 46: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

46

Noisy COMP

x 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 0 0 0 0 00 0 0 0 1 01 1 x2 0 1 x3 0 1 x7

(a) 1 0 → 1 (b) 1 0 → 1 (c) 1 0 → 01 1 1 1 0 10 0 0 0 0 00 1 1 1 1 1

Page 47: Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth

47

Noisy COMP

x 0 0 1 0 0 0 1 0 0

    M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 0 0 0 0 00 0 0 0 1 01 1 x2 0 1 x3 0 1 x7

(a) 1 0 → 1 (b) 1 0 → 1 (c) 1 0 → 01 1 1 1 0 10 0 0 0 0 00 1 1 1 1 1


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